Finite element analysis and research on the influence law of two-cell photoacoustic spectroscopy parameters

As a high-resolution labelless optical imaging technology, photoacoustic detection combines the high contrast of optical imaging and the depth of acoustic imaging, which can measure the morphology of biological particles in a natural state without damage, and has developed rapidly in recent years.¹ The essence of photoacoustic imaging is the photoacoustic effect. The pulsed laser irradiates biological particles. The temperature of the biological particles increases due to the absorption of light energy, and the volume expands and shrinks to generate sound waves.² By analyzing the sound waves, information about the biological particles is obtained.³ The frequency domain sound pressure level curve of a single cell can be used to evaluate the size and morphology of the cell, as a promising means for the early diagnosis of related diseases.⁴ The hemoglobin in red blood cells, with its high light absorption coefficient, becomes an ideal endogenous contrast agent for in vivo photoacoustic imaging.⁵ According to the frequency domain sound pressure level curve of a single red blood cell, information such as the size and shape of the red blood cell can be obtained. There is a close relationship between the shape and function of cells, so the photoacoustic measurement of cell and organelle morphology provides important information for understanding the cell function and disease diagnosis.^6,7

At present, by detecting the frequency domain sound pressure level curve of a single cell, the size and shape of a single red blood cell can be judged, and the qualitative discrimination between normal cells and abnormal cells can be performed.⁸ In another study, red blood cell aggregates were studied, and the parameters of photoacoustic signals were used to assess the degree of red blood cell aggregation in human blood.⁹ However, the former research object is a single cell; the latter research object is cell aggregates; the content of the discussion is the relationship between the photoacoustic signal and the degree of cell aggregation and does not pay attention to the mutual influence of photoacoustic signals between cells. Therefore, the effect of one cell on another cell at different distances has not been paid attention.

The existence of normal mature red blood cells in human blood is a free single cell, but the distance between cells is close; the signals will affect and interfere with each other; and the distance between cells is random. When multiple cells are tested simultaneously in vitro, the distance between cells is relatively close. In a study, a photoacoustic method was proposed to quickly quantify the morphological changes of single red blood cells. This method is more accurate than electrical impedance and light scattering methods, and faster and simpler than blood smears and optical interference methods. This study shows that red blood cells of different sizes, shapes, orientations, and compositions exhibit unique periodic minimum and maximum values in the photoacoustic spectrum above 100 MHz. However, the study only mentioned the potential of photoacoustic imaging for measuring a large number of samples and did not further analyze the detection of single cells under multi-cell conditions. The purpose of this article is to supplement and expand the content of single cell photoacoustic detection, discuss the influence of distance on cell photoacoustic signals, and find the minimum distance between cells.

When the pulsed laser is irradiated on the cell, the temperature of the cell increases due to the absorption of light energy, and the volume expands and shrinks, resulting in an initial sound pressure distribution. The sound pressure distribution p(r,t) on the cell obeys the wave equation of sound waves,

where c is the speed of sound, C_p is the specific heat capacity, is the equal pressure expansion coefficient, and H(r, t) is the heat source function.

When the pulse laser is irradiated on the target, the temperature of the target changes, the volume shrinks and expands, and the pressure changes. The temperature change and density change of the target affect the pressure change. The relationship between the photoacoustic pressure and density change and temperature change can be expressed as

where ρ is the density of the target, ρ′(r,t) is the amount of change in density, r is the position, and t is the time. The derivation of t is as follows:

Substituting $∂ρ′(r,t)∂t=−∇⋅(ρu(r,t))$ into the above formula, where u(r,t) is the vibration velocity of the particle, we can obtain

Taking the derivative of t and substituting in $ρ∂u(r,t)∂t=−∇P(r,t)$⁠, the relationship between the temperature transformation and photoacoustic pressure can be obtained as

In photoacoustic testing, the pulse is very short. At this time, the volume change caused by the temperature increase can be ignored, and the relationship between the temperature and heat source function satisfies $∂T(r,t)∂t=H(r,t)ρCp$⁠.

Thus, the relationship between the photoacoustic pressure and light energy deposition can be obtained as

The distribution characteristic of the cell photoacoustic signal in the frequency domain is that the main information and energy are concentrated in the main frequency range. Therefore, in this article, the photoacoustic signal of the same two cells is regarded as two identical point sources. Superimpose each other to satisfy the principle of acoustic interference.¹⁰ 

When two rows of sound waves with the same frequency and fixed phase difference are superimposed on each other, the corresponding relationship is as follows:

Assume that the two rows of sound waves arriving at a certain position in space are p₁ and p₂, respectively,¹¹ 

and suppose that the phase difference between two rows of sound waves arriving at this position does not change with time, but Ψ = φ₂ − φ₁ will change with position.

According to the superposition principle, the sound pressure of the synthesized sound field can be obtained as

In the above formula,

The purpose of this article is to minimize the mutual interference of photoacoustic signals between cells by adjusting the distance between the cells under the condition that the photoacoustic signals of two cells affect each other. In this paper, the finite element analysis is carried out on the influence law of the two-cell photoacoustic spectrum parameters. The research established a single cell signal model and a two-cell signal model with different distances. In the single cell model, only one red blood cell is located in the center of the spherical water environment; in the two-cell model, one red blood cell is located in the center of the spherical water environment, which is called the subject red blood cell, and the other same cell is located directly above it. Use a frequency of 400 MHz to calculate the unit wavelength, and set the distance between the upper red blood cell and the subject red blood cell according to the multiple of the wavelength, so that the simulation model of the subject red blood cell and another cell at different distances can be obtained. Using the two-cell model, we calculated the frequency domain sound pressure level curves of the subject’s red blood cells at different angles; using the single cell model, we calculated the frequency domain sound pressure level curves of a single red blood cell at different angles. The cells in the single cell model have no interference from the upper red blood cells.

The methods of judging the curve similarity mainly include the eigenvalue method and similarity function definition method. The eigenvalue method measures the similarity of two curves by comparing the characteristic parameters of the curves, while the similarity function definition method measures the similarity of two curves by comparing the distance between the two curves. The similarity function definition method is better than the eigenvalue method in comparing the similarity of the shape of two curves. When the detection point is at the same position, but the cell distance is different, the frequency domain sound pressure level curve amplitude of red blood cells is not much different. The difference is mainly reflected in the slope of the curve, so this paper selects the discrete Frechet distance based on the similarity of the shape judgment curve method. Based on the discrete Frechet distance method, taking the 75° position of the subject cell as an example, the similarity between the frequency domain sound pressure level curves is measured by the Frechet value, and it is assumed that the smaller the Frechet value, the higher the similarity.¹² The results show that the curve with the smallest Frechet value is the same curve as the most similar curve observed, that is, the Frechet value can be a good measure of the similarity of the frequency domain sound pressure level curve.

This paper uses COMSOL Multiphysics to realize the finite element analysis of cells, and the model calculates the photoacoustic signals of two red blood cells at different distances. The research established an axisymmetric model, and using the combination and segmentation of basic graphics, a biconcave red blood cell geometric model was obtained.^13–15 At the same time, the pressure acoustic transient physics was added, and a global Cartesian coordinate system was established.^16,17

The shape of a normal red blood cell is a biconcave round pie with a slight concave on both sides in the middle and thick edges. The average diameter of red blood cells is about 7.8 μm, the height is 1 μm–2 μm, and the volume is 94 μm³. The theoretical model in this paper is extended from the model developed by Evans and Fung¹⁸ (a single biconcave cell is located in a spherical system with a radius of 10 μm). According to the geometric and physical parameters of the red blood cells, the geometric shape of the red blood cell model was determined. The diameter of the red blood cells in the model was determined to be 7.82 µm, and the biconcave cells were in a circular water environment. In order to perform finite element analysis, the geometric shape of the cell model is divided into triangular meshes, and the mesh size is a standard size.

In previous studies, the wavelength of the laser beam for the best light absorption of red blood cells has been found. In blood samples, red blood cells can be regarded as the main absorber of incident light and the main radiation source of photoacoustic signals.¹⁹ Therefore, this study set the cell and water environment to an ideal state, only used red blood cells as the absorber of incident light radiation and the source of photoacoustic signal radiation, and ignored the absorption of incident light by peripheral water and the radiation outside the boundary of photoacoustic signals, and the red blood cells uniformly absorbed the laser light. Before the laser beam is irradiated, the initial pressure value of the cells is 0 Pa and the initial pressure value of water is 0 Pa. After the laser beam is irradiated, the initial pressure value of the cells is 1 Pa, and the initial pressure value of the water is 0 Pa. The mass density and sound velocity in red blood cells are 1110 kg/m³ and 1650 m/s, respectively, which are consistent with the data measured in the previous study.^20,21 The mass density and sound velocity of the water environment around the cell are 1000 kg/m³ and 1520 m/s, respectively. The model parameter settings are shown in Table I. The detection range in the time domain is 0 ns–10 ns, and the detection range in the frequency domain is 0M–1000M. The data are generated by COMSOL Multiphysics, and the post-processing of the data is completed by MATLAB R2014a.

In the two-dimensional plan view of the simulation model, a rectangular coordinate system is established, and a horizontal x axis and a vertical y axis are set. The x axis is the direction of the cell’s long diameter, and the y axis is the thickness of the cell. With the (0,0) point as the center and 90 µm as the radius, a water environment is established, and the red blood cells are in the water environment. The size and orientation of red blood cells are factors that affect the spectral characteristics of a single red blood cell. Therefore, a biconcave red blood cell with a diameter of 7.82 µm is selected to be placed in the center of the circular water environment, and the center of the red blood cell and the center of the circular water environment overlap each other. The x axis and y axis are the symmetry axes of the subject cell. As shown in Fig. 1(a), the biconcave red blood cell is in the center of the circular water environment, and the detection point is set on the right side of the cell.

Figure 1(b) is an enlarged view of the cell part in the water environment. The polar coordinate system is established with the cell center as the center. With a radius of 10 µm and the center of the cell as the center, multiple detection points are arranged on the right side of the cell to detect the signal of a single red blood cell. The positions of the detection points are −75°, −45°, 0°, 45°, 75°. Sound pressure is the change in atmospheric pressure caused by sound wave disturbance. It is the pressure change caused by a sound wave disturbance superimposed on the atmospheric pressure. The sound pressure can describe the propagation of photoacoustic signals. The sound pressure level is defined as the ratio of the sound pressure of the measured photoacoustic signal to the reference sound pressure, taking the logarithm based on 10 and multiplying by 20, and the unit is decibel (dB).

The frequency domain sound pressure level distribution of the cell reflects the morphological characteristics of the cell by periodically changing the maximum and minimum values.²² Therefore, the time-domain sound pressure signals collected from different angles are transformed and processed to obtain the frequency domain sound pressure level distribution.²³ Figure 2 is the frequency domain sound pressure level distribution of a single red blood cell at −45°, −75°, 0°, 45°, and 75°.

Here, we only analyze the frequency domain sound pressure level curve distribution from 0° to 75° and find the following characteristics: First, from 0° to 75°, the position where the minimum value begins to appear moves backward on the x axis, distributed sequentially between 200 MHz and 800 MHz. In the direction of 0°, starting from about 200 MHz, there are periodic minimum and maximum values; in the direction of 75°, the entire spectrum only has a minimum value of around 800 MHz; in the direction of 45°, starting from about 350 MHz, there are periodic minimum and maximum values. Second, the distance between the minima of each curve is increasing. In the photoacoustic spectrum of a single cell, the 0° position has the most number of minimum points in the graph, and the 75° position has the least number of minimum points in the graph. Third, before the first minimum value (200 MHz) of these curves appears, the amplitude of the curves keeps increasing from 0° to 75°.

On the basis of the single red blood cell model, the red blood cell in the center is regarded as the subject red blood cell, and another red blood cell is added above it. In this study, in order to make the distance between cells the only factor that affects the photoacoustic characteristics of the subject red blood cell, the size and orientation of the other red blood cell must be set to the same state as the subject red blood cell. The subject red blood cell is located at the center of the circular water environment, and the other red blood cell is located directly above the main red blood cell, with its center on the y axis. The position of the subject’s red blood cells in the water environment is fixed. By controlling the position of another cell on the y axis, the distance between the two cells is adjusted, and the distance between the two cells is the distance between the centers of the two cells. As shown in Fig. 3(a), it is a model diagram when the distance between two cells is 22.8 µm in a circular water environment.

The essence is the principle. The principle is that two identical wave sources are superimposed on each other. In the propagation area, a stable strengthening zone and a stable weakening zone are formed, and the strengthening zone and the weakening zone are alternately distributed. This principle is used to adjust the distance between two cells, and set the detection point in the strengthening zone as much as possible, so as to minimize interference. This principle is used to adjust the distance between two cells so that the detection points are set in the enhanced area as much as possible to minimize the interference of the detected photoacoustic signal. Observing Fig. 2, we can find that when the frequency is 400 MHz, the spectral amplitude of almost every angle is at the maximum or near the maximum. It also shows that the photoacoustic spectrum information of red blood cells is the most abundant at 400 MHz. In the model, the photoacoustic signal of the cell propagates in water at 37 °C, and the sound velocity of water at 37 °C is 1520 m/s. According to the relationship between the frequency and wavelength, the wavelength of 400 MHz is 3.8 µm. With 3.8 µm as the unit wavelength, set 11 groups of different distances, respectively, 2.85 µm (three-quarter wavelength), 3.8 µm, 7.6 µm, 11.4 µm, 15.2 µm, 19 µm, 22.8 µm, 26.6 µm, 30.4 µm, 34.2 µm, and 38 µm. Similarly, the setting of the detection point of the two-cell model is the same as the setting of the detection point of a single red blood cell. With a radius of 10 µm, the detection points are arranged at −75°, −45°, 0°, 45°, and 75° on the right side of the cell. As shown in Fig. 3(b), the relevant settings of two cells at a distance of 22.8 µm are described. In this way, the photoacoustic signal of the detection point of the subject’s red blood cell at various angles can be obtained when the subject’s red blood cell is at a different distance from another red blood cell.

Taking the frequency domain sound pressure level curve at the 75° position of the subject red blood cell as an example, the process of finding the most similar curve to a single red blood cell at 11 different distances was demonstrated, and the effects were compared. Figure 4(a) is the result of comparing the frequency domain sound pressure level curve of a single red blood cell with the frequency domain sound pressure level curve of the subject red blood cell. The red curve is the signal of a single cell at 75°, and the other curves are signals with the detection point at 75° and the distance between the subject red blood cell and another cell is three-quarters to four times the wavelength. Figure 4(b) is the result of comparing the frequency domain sound pressure level curve of a single red blood cell with the frequency domain sound pressure level curve of the subject red blood cell. The red curve is the signal of a single cell at the position of 75°, the other curves are the signal with the detection point at the position of 75°, and the distance between the subject red blood cell and another cell is five to ten times the wavelength. Comparing Figs. 4(a) and 4(b), it can be found that the curve in Fig. 4(b) is more similar to the red single cell curve, indicating that it is compared with the distance from three quarters to four times the wavelength. When the distance between cells is five times to ten times the wavelength, the influence of the signal between cells decreases, and it also shows that the distance between the cells with the smallest mutual influence is more likely to be distributed at the distance of five times to ten times the wavelength.

When performing a quantitative analysis of the frequency domain sound pressure level curve, the main criteria are the amplitude and slope of the curve. According to Figs. 4(a) and 4(b), the curve at a distance of 26.6 µm has the highest similarity to that of a single red blood cell. Both the amplitude and the slope have extremely high similarity. The two curves almost overlap. As shown in Fig. 4(c), it is the result of the comparison of the curves. One curve is the subject red blood cell at the 75° position and the distance is 26.6 µm, and the other curve is the single red blood cell at the 75° position.

Figure 4(d) shows the Frechet value at different distances. When the detection point of the subject’s red blood cell is at 75°, the frequency domain sound pressure level curve at different distances is compared with the frequency domain sound pressure level curve of a single red blood cell at 75°, and the Frechet value is calculated. At this time, it is assumed that the smaller the Frechet value, the higher the similarity. By observing the similarity between the curve of a single red blood cell at 75° and the curve of the subject red blood cell at 75° at different distances, it is judged whether the hypothesis is correct. From Fig. 4(d), it can be seen that at a distance of 26.6 µm, the Frechet value is the smallest. The results show that the distance with the smallest Frechet value is the distance with the highest curve similarity, and the Frechet value can be a good measure of the similarity of photoacoustic curves.

In order to find the distance with the least mutual influence among the 11 distances, compare the mean value of the Frechet value of each distance. In the two-cell model, there are 11 different distances between the subject red blood cell and another same cell. Under each distance, there are five different angle detection positions. Set the frequency domain sound pressure level curve of five angles at each distance as a group, calculate the Frechet value (compare the frequency domain sound pressure level curve of five angles with the frequency domain sound pressure level curve of a single red blood cell at the same position), and calculate the average of the Frechet values of the five angles in each group. As shown in Fig. 5, it is the average value of the Frechet values of 11 distances of the subject red blood cells. As the distance increases, the overall Frechet average value shows a trend of first decline and then rise, reaching the minimum value at 26.6 µm. That is, as the distance increases, the similarity between the frequency domain sound pressure level curve of the subject red blood cell and the frequency domain sound pressure level curve of a single red blood cell at the same position first increases and then decreases, and the similarity reaches the highest at 26.6 µm. It can be seen that in the two-cell model, when the two cells are at 26.6 µm, the photoacoustic signal interaction between the cells is the least.

In order to verify that 26.6 µm is the distance where the photoacoustic signals of the two cells have the least influence on each other, the five positions of the subject’s red blood cells performed the same operation as the above 75° position. That is, for each position, compare the frequency domain sound pressure level curve of 11 different distances of the subject red blood cell with the frequency domain sound pressure level curve of a single red blood cell at the same position and find the curve with the smallest Frechet value among the 11 distances. When the detection point is at −75°, −45°, 45°, and 75°, the minimum value of the Frechet value appears at 26.6 µm, and the minimum value of the Frechet value at the 0° position appears at 34.2 µm, but the Frechet value at 26.6 µm is only three units away from the value at 34.2 µm, as shown in Fig. 6(c), and when the detection point is at 0°, the curve at 26.6 µm still has a high similarity to a single red blood cell. It can be approximated that the curve that is most similar to the single cell curve at 0° appears at 26.6 µm. Observing Figs. 6(a)–6(e), it is the comparison result of the subject red blood cells at −75°, −45°, 0°, 45°, and 75° when the distance between the two cells is 26.6 µm. Observing Fig. 3, it can be seen that when the distance between the two cells is 26.6 µm, the sound pressure level curve of the subject red blood cell at each angle is very similar to the unaffected single cell sound pressure level curve. The results verify the above judgment, that is, when the distance between the two cells is 26.6 µm, the mutual influence of the photoacoustic signals between the cells is minimal.

Experiments and finite element simulations have proved that the photoacoustic power spectrum of a single red blood cell has good sensitivity to the morphological characteristics of the cell, but there are still obstacles in achieving simultaneous detection of multi-cell photoacoustic signals. Under multicellular conditions, the interaction of photoacoustic signals between cells has not been discussed. This article finds the optimal distance between two identical red blood cells, at which the photoacoustic signals between the two cells interfere with each other the least. The photoacoustic spectrum information of a single red blood cell is the most abundant at 400 MHz. The unit wavelength of 400 MHz is calculated as 3.8 µm. The range of the distance between two cells is three-quarters to ten times the unit wavelength. The results of the study show that within ten times the wavelength distance, as the distance increases, the interference between cells first decreases and then increases, and at seven times the unit wavelength, the interference between cells is the smallest. This is the result of sound wave interference. Two sound sources emit sound waves. When the phases are the same, the amplitude of the two waves increases and the sound pressure amplitude is strengthened.

However, this conclusion needs to be confirmed by experiments because, in actual operation, there will be other factors that affect the results of the experiment, and all factors need to be included to make the conclusion more consistent with the actual situation. In the previous study, the photoacoustic spectrum was quantitatively analyzed. The method used was to analyze the amplitude of a certain frequency and the slope of a certain frequency in the photoacoustic spectrum. Only part of the information of the photoacoustic spectrum was extracted, and it is suitable for situations where there is a large difference between the shapes of the cells to be detected. In order to further quantitatively analyze the cell photoacoustic spectrum, this paper introduces the Frechet distance to measure the similarity of the curves, quantifies the subtle differences between the curves, analyzes all or part of the frequency range, and uses the Frechet distance as a powerful tool to analyze the photoacoustic curve. This article uses two identical double-concave red blood cells as an example. The model calculates the similarity between the frequency domain sound pressure level curve of the subject red blood cell (at different distances) and the frequency domain sound pressure level curve of an unaffected single red blood cell. Get the distance with the highest similarity, that is, the distance with the least mutual influence. The application of this method is not limited to the characterization of red blood cells. Almost any cell containing endogenous contrast agent and exogenous contrast agent can find the distance between cells with the least interference by this method. This article supplements the content of detecting photoacoustic signals of multiple cells in photoacoustic microscopy.

In previous studies, photoacoustic rapid quantitative analysis of the morphology of a single red blood cell has been achieved, but there is no further discussion on how to detect the photoacoustic signal of a single cell under multi-cell conditions, that is, the interaction between cell signals is not considered. For this reason, this article discusses the influence of photoacoustic signals between two red blood cells at different distances. The study carried out the finite element analysis on the influence law of two-cell photoacoustic spectrum parameters and established a single cell signal model and a two-cell signal model with different distances. Using the two-cell signal model, the frequency domain sound pressure level curves of the subject red blood cell at different distances and angles were calculated; using the single cell signal model, the frequency domain sound pressure level curves of a single red blood cell at different angles were calculated. Compare the frequency domain sound pressure level curve of the subject red blood cell at five angles and 11 distances with the frequency domain sound pressure level curve of a single red blood cell at the same angle. The Frechet value is used to measure the similarity of the photoacoustic curve. The smaller the Frechet value, the more similar the two curves. Among them, the calculation of the mutual influence of the signals between cells and the use of the similarity function definition method to measure the similarity of the photoacoustic curves have not been discussed in previous studies.

It is observed that the mean value of the Frechet value decreases first and then rises as the distance increases, and reaches the minimum value at 26.6 µm. In addition, when the distance between the two cells is 26.6 µm, the frequency domain sound pressure level curves of the subject red blood cells at five angles are compared with the frequency domain sound pressure level curves of the unaffected single cells at the same position. It is found that the Frechet values of the five angles also reach the minimum or close to the minimum at 26.6 µm, that is, at a distance of 26.6 µm, The frequency domain sound pressure level curve of the subject red blood cell at various angles has the highest similarity with the frequency domain sound pressure level curve of a single cell. These data indicate that when two biconcave red blood cells are at a distance of 26.6 µm, the mutual influence of photoacoustic signals between the cells is minimal. Future work will combine other methods on this basis to achieve further separation and detection of multi-cell photoacoustic signals.

This project was sponsored in part by the National Natural Science Foundation of China (Grant No. 61671414) and the China Postdoctoral Science Foundation (Grant No. 2017M611198).

The data that support the findings of this study are available within the article.